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Decay rates for the damped wave equation on the torus
Nalini Anantharaman, Matthieu Léautaud, Stéphane Nonnenmacher
To cite this version:
Nalini Anantharaman, Matthieu Léautaud, Stéphane Nonnenmacher. Decay rates for the damped
wave equation on the torus. 2012. �hal-00740492�
Decay rates for the damped wave equation on the torus
With an appendix by St´
ephane Nonnenmacher
∗
Nalini Anantharaman
†and Matthieu L´eautaud
‡,
Universit´e Paris-Sud 11, Math´ematiques, Bˆatiment 425, 91405 Orsay Cedex, France
October 10, 2012
Abstract
We address the decay rates of the energy for the damped wave equation when the damping coefficient b does not satisfy the Geometric Control Condition (GCC). First, we give a link with the controllability of the associated Schr¨odinger equation. We prove in an abstract setting that the observability of the Schr¨odinger group implies that the semigroup associated to the damped wave equation decays at rate 1/√t (which is a stronger rate than the general logarithmic one predicted by the Lebeau Theorem).
Second, we focus on the 2-dimensional torus. We prove that the best decay one can expect is 1/t, as soon as the damping region does not satisfy GCC. Conversely, for smooth damping coefficients b, we show that the semigroup decays at rate 1/t1−ε, for all ε > 0. The proof relies
on a second microlocalization around trapped directions, and resolvent estimates.
In the case where the damping coefficient is a characteristic function of a strip (hence dis-continuous), St´ephane Nonnenmacher computes in an appendix part of the spectrum of the associated damped wave operator, proving that the semigroup cannot decay faster than 1/t2/3.
In particular, our study shows that the decay rate highly depends on the way b vanishes.
Keywords
Damped wave equation, polynomial decay, observability, Schr¨odinger group, torus, two-microlocal semiclassical measures, spectrum of the damped wave operator.
Contents
I
The damped wave equation
2
1 Decay of energy: a survey of existing results 2
2 Main results of the paper 5
2.1 The damped wave equation in an abstract setting . . . 5 2.2 Decay rates for the damped wave equation on the torus . . . 7 2.3 Some related open questions . . . 9
II
Resolvent estimates and stabilization in the abstract setting
10
3 Proof of Theorem 2.3 assuming Proposition 2.4 10
∗
4 Proof of Proposition 2.4 10
III
Proof of Theorem 2.6: smooth damping coefficients on the torus
15
5 Semiclassical measures 16
6 Zero-th and first order informations on µ 17 7 Geometry on the torus and decomposition of invariant measures 19 7.1 Resonant and non-resonant vectors on the torus . . . 19 7.2 Decomposition of invariant measures . . . 20 7.3 Geometry of the subtori TΛ and TΛ⊥ . . . 21
8 Change of quasimode and construction of an invariant cutoff function 22 9 Second microlocalization on a resonant affine subspace 23 10 Propagation laws for the two-microlocal measures νΛ and ρ
Λ 28
10.1 Propagation of νΛ . . . 28
10.2 Propagation of ρΛ . . . 30
11 The measures νΛ and ρ
Λ vanish identically. End of the proof of Theorem 2.6 32
12 Proof of Proposition 8.2 33
13 Proof of Proposition 8.3: existence of the cutoff function 35
IV
An a priori lower bound for decay rates on the torus: proof of
Theorem 2.5
41
A Pseudodifferential calculus 42
B Spectrum of P (z) for a piecewise constant damping
(by St´ephane Nonnenmacher) 42
Part I
The damped wave equation
1
Decay of energy: a survey of existing results
Let (M, g) be a smooth compact connected Riemannian d-dimensional manifold with or without boundary ∂M . We denote by ∆ the (non-positive) Laplace-Beltrami operator on M for the metric g. Given a bounded nonnegative function, b∈ L∞(M ), b(x)≥ 0 on M, we want to understand the
asymptotic behaviour as t→ +∞ of the solution u of the problem ∂2 tu− ∆u + b(x)∂tu = 0 in R+× M, u = 0 on R+ × ∂M (if ∂M 6= ∅), (u, ∂tu)|t=0= (u0, u1) in M. (1.1)
The energy of a solution is defined by E(u, t) = 1
2(k∇u(t)k
2
Multiplying (1.1) by ∂tu and integrating on M yields the following dissipation identity d dtE(u, t) =− Z M b|∂tu|2dx,
which, as b is nonnegative, implies a decay of the energy. As soon as b≥ C > 0 on a nonempty open subset of M , the decay is strict and E(u, t)→ 0 as t → +∞. The question is then to know at which rate the energy goes to zero.
The first interesting issue concerns uniform stabilization: under which condition does there exist a function F (t), F (t)→ 0, such that
E(u, t)≤ F (t)E(u, 0) ? (1.3) The answer was given by Rauch and Taylor [RT74] in the case ∂M =∅ and by Bardos, Lebeau and Rauch [BLR92] in the general case (see also [BG97] for the necessity of this condition): assuming that b∈ C0(M ), uniform stabilisation occurs if and only if the set
{b > 0} satisfies the Geometric Control Condition (GCC). Recall that a set ω ⊂ M is said to satisfy GCC if there exists L0 > 0
such that every geodesic γ (resp. generalised geodesic in the case ∂M 6= ∅) of M with length larger than L0 satisfies γ∩ ω 6= ∅. Under this condition, one can take F (t) = Ce−κt (for some constants
C, κ > 0) in (1.3), and the energy decays exponentially. Finally, Lebeau gives in [Leb96] the explicit (and optimal) value of the best decay rate κ in terms of the spectral abscissa of the generator of the semigroup and the mean value of the function b along the rays of geometrical optics.
In the case where {b > 0} does not satisfy GCC, i.e. in the presence of “trapped rays” that do not meet{b > 0}, what can be said about the decay rate of the energy? As soon as b ≥ C > 0 on a nonempty open subset of M , Lebeau shows in [Leb96] that the energy (of smoother initial data) goes at least logarithmically to zero (see also [Bur98]):
E(u, t)≤ C f(t)2ku0k2H2(M)∩H1 0(M)+ku1k 2 H1 0(M) , for all t > 0, (1.4) with f (t) = log(2+t)1 (where H2(M )∩H1
0(M ) and H01(M ) have to be replaced by H2(M ) and H1(M )
respectively if ∂M =∅). Note that here, f(t)2 characterizes the decay of the energy, whereas f (t) is that of the associated semigroup. Moreover, the author constructed a series of explicit examples of geometries for which this rate is optimal, including for instance the case where M = S2 is the
two-dimensional sphere and {b > 0} ∩ Nε =∅, where Nε is a neighbourhood of an equator of S2.
This result is generalised in [LR97] for a wave equation damped on a (small) part of the boundary. In this paper, the authors also make the following comment about the result they obtain:
“Notons toutefois qu’une ´etude plus approfondie de la localisation spectrale et des taux de d´ecroissance de l’´energie pour des donn´ees r´eguli`eres doit faire intervenir la dynamique globale du flot g´eod´esique g´en´eralis´e sur M . Les th´eor`emes [LR97, Th´eor`eme 1] et [LR97, Th´eor`eme 2] ne four-nissent donc que les bornes a priori qu’on peut obtenir sans aucune hypoth`ese sur la dynamique, en n’utilisant que les in´egalit´es de Carleman qui traduisent “l’effet tunnel”.”
In all examples where the optimal decay rate is logarithmic, the trapped ray is a stable trajectory from the point of view of the dynamics of the geodesic flow. This means basically that an important amount of the energy can stay concentrated, for a long time, in a neighbourhood of the trapped ray, i.e. away from the damping region.
If the trapped trajectories are less stable, or unstable, one can expect to obtain an intermediate decay rate, between exponential and logarithmic. We shall say that the energy decays at rate f (t) if (1.4) is satisfied (more generally, see Definition 2.2 below in the abstract setting). This problem has already been adressed and, in some particular geometries, several different behaviours have been exhibited. Two main directions have been investigated.
On the one hand, Liu and Rao considered in [LR05] the case where M is a square and the set {b > 0} contains a vertical strip. In this situation, the trapped trajectories consist in a family
of parallel vertical geodesics; these are unstable, in the sense that nearby geodesics diverge at a linear rate. They proved that the energy decays at ratelog(t)t
1 2
(i.e., that (1.4) is satisfied with f (t) = log(t)t
1 2
). This was extended by Burq and Hitrik [BH07] (see also [Nis09]) to the case of partially rectangular two-dimensional domains, if the set {b > 0} contains a neighbourhood of the non-rectangular part. In [Phu07], Phung proved a decay at rate t−δ for some (unprecised) δ > 0 in
a three-dimensional domain having two parallel faces. In all these situations, the only obstruction to GCC is due to a “cylinder of periodic orbits”. The geometry is flat and the unstabilities of the geodesic flow around the trapped rays are relatively weak (geodesics diverge at a linear rate).
In [BH07], the authors argue that the optimal decay in their geometry should be of the form t1−ε1 ,
for all ε > 0. They provide conditions on the damping coefficient b(x) under which one can obtain such decay rates, and wonder whether this is true in general. Our main theorem (see Theorem 2.6 below) extends these results to more general damping functions b on the two-dimensional torus.
On the other hand, Christianson [Chr10] proved that the energy decays at rate e−C√t for some
C > 0, in the case where the trapped set is a hyperbolic closed geodesic. Schenck [Sch11] proved an energy decay at rate e−Cton manifolds with negative sectional curvature, if the trapped set is “small
enough” in terms of topological pressure (for instance, a small neighbourhood of a closed geodesic), and if the damping is “large enough” (that is, starting from a damping function b, βb will work for any β > 0 sufficiently large). In these two papers, the geodesic flow near the trapped set enjoys strong instability properties: the flow on the trapped set is uniformly hyperbolic, in particular all trajectories are exponentially unstable.
These cases confirm the idea that the decay rates of the energy strongly depends on the stability of trapped trajectories.
One may now want to compare these geometric situations to situations where the Schr¨odinger group is observable (or, equivalently, controllable), i.e. for which there exist C > 0 and T > 0 such that, for all u0∈ L2(M ), we have
ku0k2L2(M)≤ C
Z T 0 k
√
b e−it∆u0k2L2(M)dt. (1.5)
The conditions under which this property holds are also known to be related to stability of the geodesic flow. In particular, the works [BLR92], [LR05], [BH07, Nis09] and [Chr10, Sch11] can be seen as counterparts for damped wave equations of the articles [Leb92], [Har89a, Jaf90], [BZ04] and [AR10], respectively, in the context of observation of the Schr¨odinger group.
Our main results are twofold. First, we clarify (in an abstract setting) the link between the ob-servability (or the controllability) of the Schr¨odinger equation and polynomial decay for the damped wave equation. This follows the spirit of [Har89b], [Mil05], exploring the links between the different equations and their control properties (e.g. observability, controllability, stabilization...). More pre-cisely, we prove that the controllability of the Schr¨odinger equation implies a polynomial decay at rate √1
t for the damped wave equation (Theorem 2.3).
Second, we study precisely the damped wave equation on the flat torus T2 in case GCC fails.
We give the following a priori lower bound on the decay rate, revisiting the argument of [BH07]: (1.1) is not stable at a better rate than 1
t, provided that GCC is not satisfied. In this situation, the
Schr¨odinger group is known to be controllable (see [Jaf90], [Kom92] and the more recent works [AM11] and [BZ11]). Thus, one cannot hope to have a decay better than polynomial in our previous result, i.e. under the mere assumption that the Schr¨odinger flow is observable.
The remainder of the paper is devoted to studying the gap between the a priori lower and upper bounds given respectively by 1
t and 1 √
t on flat tori. For smooth nonvanishing damping coefficient
b(x), we prove that the energy decays at rate 1
t1−ε for all ε > 0. This result holds without making
any dynamical assumption on the damping coefficient, but only on the order of vanishing of b. It generalises a result of [BH07], which holds in the case where b is invariant in one direction. Our
analysis is, again, inspired by the recent microlocal approach proposed in [AM11] and [BZ11] for the observability of the Schr¨odinger group. More precisely, we follow here several ideas and tools introduced in [Mac10] and [AM11].
In the situation where b is a characteristic function of a vertical strip of the torus (hence discon-tinuous), St´ephane Nonnenmacher proves in Appendix B that the decay rate cannot be faster than
1
t2/3. This is done by explicitly computing the high frequency eigenvalues of the damped wave
oper-ator which are closest to the imaginary axis (see for instance the figures in [AL03, AL12]). The fact that the decay rate 1/t is not achieved in this situation was observed in the numerical computations presented in [AL12].
In contrast to the control problem for the Sch¨odinger equation, this result shows that the stabi-lization of the wave equation is not only sensitive to the global properties of the geodesic flow, but also to the rate at which the damping function vanishes.
2
Main results of the paper
Our first result can be stated in a general abstract setting that we now introduce. We come back to the case of the torus afterwards.
2.1
The damped wave equation in an abstract setting
Let H and Y be two Hilbert spaces (resp. the state space and the observation/control space) with normsk · kH andk · kY, and associated inner products (·, ·)H and (·, ·)Y.
We denote by A : D(A)⊂ H → H a nonnegative selfadjoint operator with compact resolvent, and B ∈ L(Y ; H) a control operator. We recall that B∗ ∈ L(H; Y ) is defined by (B∗h, y)
Y = (h, By)H
for all h∈ H and y ∈ Y .
Definition 2.1. We say that the system
∂tu + iAu = 0, y = B∗u, (2.1)
is observable in time T if there exists a constant KT > 0 such that, for all solution of (2.1), we have
ku(0)k2 H≤ KT Z T 0 ky(t)k 2 Ydt.
We recall that the observability of (2.1) in time T is equivalent to the exact controllability in time T of the adjoint problem
∂tu + iAu = Bf, u(0) = u0, (2.2)
(see for instance [Leb92] or [RTTT05]). More precisely, given T > 0, the exact controllability in time T is the ability of finding for any u0, u1∈ H a control function f ∈ L2(0, T ; Y ) so that the solution
of (2.2) satisfies u(T ) = u1.
We equipH = D(A12)× H with the graph norm
k(u0, u1)k2H=k(A + Id)
1 2u
0k2H+ku1k2H,
and define the seminorm
|(u0, u1)|2H=kA
1 2u
0k2H+ku1k2H.
Of course, if A is coercive on H,| · |H is a norm on H equivalent to k · kH.
We also introduce in this abstract setting the damped wave equation on the spaceH, (
∂2
tu + Au + BB∗∂tu = 0,
(u, ∂tu)|t=0= (u0, u1)∈ H,
which can be recast onH as a first order system ∂tU =AU, U|t=0=t(u0, u1), U = u ∂tu , A = 0 Id −A −BB∗ , D(A) = D(A) × D(A12). (2.4)
The compact injections D(A) ֒→ D(A12) ֒→ H imply that D(A) ֒→ H compactly, and that the
operatorA has a compact resolvent.
We define the energy of solutions of (2.3) by E(u, t) = 1 2 kA 1 2uk2 H+k∂tuk2H =1 2|(u, ∂tu)| 2 H2.
Definition 2.2. Let f be a function such that f (t)→ 0 when t → +∞. We say that System (2.3) is stable at rate f (t) if there exists a constant C > 0 such that for all (u0, u1)∈ D(A), we have
E(u, t)12 ≤ Cf(t)|A(u0, u1)|
H, for all t > 0.
If it is the case, for all k > 0, there exists a constant Ck > 0 such that for all (u0, u1)∈ D(Ak), we
have (see for instance [BD08, page 767])
E(u, t)12 ≤ Ck f (t)kkAk(u0, u1)k
H, for all t > 0.
Theorem 2.3. Suppose that there exists T > 0 such that System (2.1) is observable in time T . Then System (2.3) is stable at rate √1
t.
Note that the gain of the log(t)12 with respect to [LR05, BH07] is not essential in our work. It is
due to the optimal characterization of polynomially decaying semigroups obtained by Borichev and Tomilov [BT10].
This Theorem may be compared with the works (both presented in a similar abstract setting) [Har89b] by Haraux, proving that the controllability of wave-type equations in some time is equivalent to uniform stabilization of (2.3), and [Mil05] by Miller, showing that the controllability of wave-type equations in some time implies the controllability of Schr¨odinger-type equations in any time.
Note that the link between this abstract setting and that of Problem (1.1) is H = Y = L2(M ),
A =−∆ with D(A) = H2(M ) if ∂M =∅ and H2(M )∩ H1
0(M ) otherwise, B is the multiplication in
L2(M ) by the bounded function√b.
As a first application of Theorem 2.3 we obtain a different proof of the polynomial decay results for wave equations of [LR05] and [BH07] as consequences of the associated control results for the Schr¨odinger equation of [Har89a] and [BZ04] respectively.
Moreover, Theorem 2.3 provides also several new stability results for System (1.1) in particu-lar geometric situations; namely, in all following situations, the Schr¨odinger group is proved to be observable, and Theorem 2.3 gives the polynomial stability at rate √1
t for (1.1):
• For any nonvanishing b(x) ≥ 0 in the 2-dimensional square (resp. torus), as a consequence of [Jaf90] (resp. [Mac10, BZ11]); for any nonvanishing b(x) ≥ 0 in the d-dimensional rectangle (resp. d-dimensional torus) as a consequence of [Kom92] (resp. [AM11]);
• If M is the Bunimovich stadium and b(x) > 0 on the neighbourhood of one half disc and on one point of the opposite side, as a consequence of [BZ04];
• If M is a d-dimensional manifold of constant negative curvature and the set of trapped tra-jectories (as a subset of S∗M , see [AR10, Theorem 2.5] for a precise definition) has Hausdorff
Moreover, Lebeau gives in [Leb96, Th´eor`eme 1 (ii)] several 2-dimensional examples for which the decay rate log(2+t)1 is optimal. For all these geometrical situations, Theorem 2.3 implies that the Schr¨odinger group is not observable.
The proof of Theorem 2.3 relies on the following characterization of polynomial decay for Sys-tem (2.3). For z ∈ C, we define on H the operator P (z) = A + z2Id +zBB∗, with domain
D(P (z)) = D(A). We prove in Lemma 4.2 below that P (is) is invertible for all s∈ R, s 6= 0. Proposition 2.4. Suppose that
for any eigenvector ϕ of A, we have B∗ϕ6= 0. (2.5)
Then, for all α > 0, the five following assertions are equivalent: The system (2.3) is stable at rate 1
tα, (2.6)
There exist C > 0 and s0≥ 0 such that for all s ∈ R, |s| ≥ s0, k(is Id −A)−1kL(H)≤ C|s|
1 α, (2.7)
There exist C > 0 and s0≥ 0 such that for all z ∈ C, satisfying |z| ≥ s0,
and| Re(z)| ≤ 1
C| Im(z)|α1, we have k(z Id −A)
−1k
L(H)≤ C| Im(z)|
1
α, (2.8)
There exist C > 0 and s0≥ 0 such that for all s ∈ R, |s| ≥ s0, kP (is)−1kL(H)≤ C|s|
1
α−1, (2.9)
There exists C > 0 and s0≥ 0 such that for all s ∈ R, |s| ≥ s0 and u∈ D(A),
kuk2 H ≤ C |s| 2 α−2kP (is)uk2 H+|s| 1 αkB∗uk2 Y . (2.10)
This proposition is proved as a consequence of the characterization of polynomial decay for general semigroups in terms of resolvent estimates given in [BT10], providing the equivalence between (2.6) and (2.7). See also [BD08] for general decay rates in Banach spaces. Note in particular that the proof of a decay rate is reduced to the proof of a resolvent estimate on the imaginary axes. By the way, this estimate implies the existence of a “spectral gap” between the spectrum ofA and the imaginary axis, given by (2.8).
Note finally that the estimates (2.7), (2.9) and (2.10) can be equivalently restricted to s > 0, since P (−is)u = P (is)u.
2.2
Decay rates for the damped wave equation on the torus
The main results of this article deal with the decay rate for Problem (1.1) on the torus T2 :=
(R/2πZ)2. In this setting, as well as in the abstract setting, we shall write P (z) =
−∆ + z2+ zb(x).
First, we give an a priori lower bound for the decay rate of the damped wave equation, on T2, when GCC is “strongly violated”, i.e. assuming that supp(b) does not satisfy GCC (instead of {b > 0}). This theorem is proved by constructing explicit quasimodes for the operator P (is). Theorem 2.5. Suppose that there exists (x0, ξ0)∈ T∗T2, ξ06= 0, such that
{b > 0} ∩ {x0+ τ ξ0, τ ∈ R} = ∅.
Then there exist two constants C > 0 and κ0> 0 such that for all n∈ N,
kP (inκ0)−1kL(L2(T2))≥ C. (2.11)
As a consequence of Proposition 2.4, polynomial stabilization at rate t1+ε1 for ε > 0 is not
possible if there is a strongly trapped ray (i.e. that does not intersect supp(b)). More precisely, in such geometry, Theorem 2.5 combined with Lemma 4.6 and [BD08, Proposition 1.3] shows that m1(t)≥1+tC , for some C > 0 (with the notation of [BD08] where m1(t) denotes the best decay rate).
Then, the main goal of this paper is to explore the gap between the a priori upper bound √1 t for
the decay rate, given by Theorem 2.3, and the a priori lower bound 1t of Theorem 2.5. Our results are twofold (somehow in two opposite directions) and concern either the case of smooth damping functions b, or the case b = 1U, with U ⊂ T2.
2.2.1 The case of smooth damping coefficients
Our main result deals with the case of smooth damping coefficients. Without any geometric assump-tion, but with an additional hypothesis on the order of vanishing of the damping function b, we prove a weak converse of Theorem 2.5.
Theorem 2.6. Let M = T2with the standard flat metric. There exists ε
0> 0 satisfying the following
property. Suppose that b is a nonnegative nonvanishing function on T2 satisfying√b
∈ C∞(T2) and
that there exist ε∈ (0, ε0) and Cε> 0 such that
|∇b(x)| ≤ Cεb1−ε(x), for x∈ T2. (2.12)
Then, there exist C > 0 and s0≥ 0 such that for all s ∈ R, |s| ≥ s0,
kP (is)−1k
L(L2(T2)) ≤ C|s|δ, with δ = 8ε (2.13)
As a consequence of Proposition 2.4, in this situation, the damped wave equation (1.1) is stable at rate 1
t1+δ1 .
Following carefully the steps of the proof, one sees that ε0 = 761 works, but the proof is not
optimized with respect to this parameter, and it is likely that it could be much improved.
One of the main difficulties in understanding the decay rates is that there exists no general monotonicity property of the type “b1(x) ≤ b2(x) for all x =⇒ the decay rate associated to the
damping b2 is larger (or smaller) than the decay rate associated to the damping b1”. This makes a
significant difference with observability or controllability problems of the type (1.5).
Assumption (2.12) is only a local assumption in a neighbourhood of ∂{b > 0} (even if it is stated here globally on T2). Far from this set, i.e. on each compact set
{b ≥ b0} for b0 > 0, the constant
Cε can be choosen uniformly, depending only on b0, and not on ε. Hence, ε somehow quantifies the
vanishing rate of the damping function b.
An interesting situation is when the smooth function b vanishes like e− 1
xα in smooth local
coordi-nates, for some α > 0. In this case, Assumption (2.12) is satisfied for any ε > 0, and the associated damped wave equation (1.1) is stable at rate t1−δ1 for any δ > 0. This shows that the lower bound
given by Theorem 2.5, as well as the decay rate 1
t, are sharp in general. This phenomenon had
already been remarked by Burq and Hitrik in [BH07] in the case where b is invariant in one direction. Typical smooth functions not satisfying Assumption (2.12) are for instance functions vanishing like sin(1x)2e−1
x. We do not have any idea of the decay rate achieved in this case (except for the a
priori bounds √1 t and
1 t).
Theorem 2.6 generalises the result of [BH07], which only holds if b is assumed to be invariant in one direction. Our proof is based on ideas and tools developped in [Mac10, AM11] and especially on two-microlocal semiclassical measures. One of the key technical points appears in Section 13: we have to construct, for each trapped direction, a cutoff function invariant in that direction and adapted to the damping coefficient b. We do not know how to adapt this technical construction to tori of higher dimension, d > 2; hence we do not know whether Theorem 2.6 holds in higher dimension (although we have no reason to suspect it should not hold). Only in the particular case where b is invariant in d− 1 directions can our methods (or those of [BH07]) be applied to prove the analogue of Theorem 2.6.
Note that if GCC is satisfied, one has (on a general compact manifold M ) for some C > 1 and all|s| ≥ s0 the estimate
kP (is)−1k
L(L2(M)) ≤ C|s|−1. (2.14)
instead of (2.13). Estimate (2.14) is in turn equivalent to uniform stabilization (see [Hua85] together with Lemma 4.6 below).
Remark 2.7. As a consequence of Theorem 2.6 on the torus, we can deduce that the decay rate t−1+δ1 also holds for Equation (1.1) if M = (0, π)2is the square, one takes with Dirichlet or Neumann
boundary conditions, and the damping function b is smooth, vanishes near ∂M and satisfies As-sumption (2.12). First, we extend the function b as an even (with respect to both variables) smooth function on the larger square (−π, π)2, and using the injection ı : (
−π, π)2
→ T2, as a smooth
func-tion on T2, still satisfying (2.12). Moreover, D(∆
D) (resp. D(∆N)) on (0, π)2 can be identified as
the closed subspace of odd (resp. even) functions of D(∆D) (resp. D(∆N)) on (−π, π)2. Using again
the injection ı, it can also be identified with a closed subspace of H2(T2). The estimate
kukL2(T2)≤ C|s|δkP (is)ukL2(T2) for all u∈ H2(T2),
is thus also true on the square (0, π)2for Dirichlet or Neumann boundary conditions. In particular,
this strongly improves the results of [LR05].
The lower bound of Theorem 2.5 can be similarly extended to the case of a square with Dirichlet or Neumann boundary conditions, implying that the rate 1t is optimal if GCC is strongly violated.
2.2.2 The case of discontinuous damping functions
Appendix B (by St´ephane Nonnenmacher) deals with the case where b is the characteristic function of a vertical strip, i.e. b = eB1U, for some eB > 0 and U = (a, b)× T ⊂ T2. Due to the invariance of b
in one direction, the spectrum of the damped wave operatorA splits into countably many “branches” of eigenvalues. This structure of the spectrum is illustrated in the numerics of [AL03, AL12].
The branch closest to the imaginary axis is explicitly computed, it contains a sequence of eigen-values (zi)i∈N such that Im zi → ∞ and | Re zi| ≤ (Im zCi0)3/2. This result is in agreement with the
numerical tests given in [AL12].
As a consequence, for any ε > 0 and C > 0, the strip | Re z| ≤ C| Im(z)|−3/2+ε contains
in-finitely many poles of the resolvent (z Id−A)−1, so item (2.8) in Proposition 2.4 implies the following
obstruction to the stability of this damped system :
Corollary 2.8. For any ε > 0, the damped wave equation (1.1) on T2 with the damping function
(B.1) cannot be stable at the rate 1 t2/3+ε.
The same result holds on the square with Dirichlet or Neumann boundary conditions.
More precisely, in this situation, Lemma 4.6 and [BD08, Proposition 1.3] yield that m1(t) ≥ C
(1+t)2/3, for some C > 0 (with the notation of [BD08] where m1(t) denotes the best decay rate).
This corollary shows in particular that the regularity conditions in Theorem 2.6 cannot be com-pletely disposed of if one wants a stability at the rate 1/t1−ε for small ε.
2.3
Some related open questions
The various results obtained in this article lead to several open questions.
1. In the case where b is the characteristic function of a vertical strip, our analysis shows that the best decay rate lies somewhere between 1
t12 and
1
t23, but the “true” decay rate is not yet clear.
2. It would also be interesting to investigate the spectrum and the decay rates for damping functions b invariant in one direction, but having a less singular behaviour than a characteristic function. In particular, is it possible to give a precise link between the vanishing rate of b and the decay rate?
3. In the general setting of Section 2.1 (as well as in the case of the damped wave equation on the torus), is the a priori upper bound 1
t12 for the decay rate optimal?
4. For smooth damping functions vanishing like e− 1
xα, Theorem 2.6 yields stability at rate 1
t1−δ
for all δ > 0. Is the decay rate 1t reached in this situation? Can one find a damping function b such that the decay rate is exactly 1t?
5. The lower bound of of Theorem 2.5 is still valid in higher dimensional tori. Is there an analogue of Theorem 2.6 (i.e. for general “smooth” damping functions) for Td, with d≥ 3?
Part II
Resolvent estimates and stabilization in
the abstract setting
3
Proof of Theorem 2.3 assuming Proposition 2.4
To prove Theorem 2.3, we express the observability condition as a resolvent estimate (also known as the Hautus test), as introduced by Burq and Zworski [BZ04], and further developed by Miller [Mil05] and Ramdani, Takahashi, Tenenbaum and Tucsnak [RTTT05]. For a survey of this notion, we refer to the book [TW09, Section 6.6].
In particular [Mil05, Theorem 5.1] (or [TW09, Theorem 6.6.1]) yields that System (2.1) is ob-servable in some time T > 0 if and only if there exists a constant C > 0 such that we have
kuk2H≤ C k(A − λ Id)uk2H+kB∗uk2Y
, for all λ∈ R and u ∈ D(A).
As a first consequence, Assumption (2.5) is satisfied and Proposition 2.4 applies in this context. Moreover, we have, for all s∈ R and u ∈ D(A),
kuk2H ≤ C k(A − s2Id +isBB∗− isBB∗)uk2H+kB∗uk2Y
≤ C kP (is)uk2H+ s2kBB∗uk2H+kB∗uk2Y
(3.1) Since B∈ L(Y ; H), we obtain for s ≥ 1 and for some C > 0,
kuk2 H≤ C kP (is)uk2H+ s2kB∗uk2Y ≤ C s2 kP (is)uk2 H+ s2kB∗uk2Y . Proposition 2.4 then yields the polynomial stability at rate √1
t for (2.3). This concludes the proof of
Theorem 2.3.
4
Proof of Proposition 2.4
Our proof strongly relies on the characterization of polynomially stable semigroups, given in [BT10, Theorem 2.4], which can be reformulated as follows.
Theorem 4.1 ([BT10], Theorem 2.4). Let (et ˙A)t≥0 be a bounded C0-semigroup on a Hilbert space
˙
H, generated by ˙A. Suppose that iR ∩ Sp( ˙A) = ∅. Then, the following conditions are equivalent: ket ˙AA˙−1kL( ˙H)=O(t−α), as t→ +∞, (4.1)
k(is Id − ˙A)−1k
L( ˙H)=O(|s|
1
α), as s→ ∞. (4.2)
Let us first describe some spectral properties of the operatorA defined in (2.4). Lemma 4.2. The spectrum ofA contains only isolated eigenvalues and we have
Sp(A) ⊂ −12kB∗k2 L(H;Y ), 0 + iR ∪[−kB∗k2 L(H;Y ), 0] + 0i , with ker(A) = ker(A) × {0}.
Moreover, the operator P (z) is an isomorphism from D(A) onto H if and only if z /∈ Sp(A). If this is satisfied, we have
(z Id−A)−1= P (z)−1(BB∗+ z Id) P (z)−1 P (z)−1(zBB∗+ z2Id) − Id zP (z)−1 . (4.3)
The localization properties for the spectrum of A, stated in the first part of this lemma are illustrated for instance in [AL03] or [AL12].
This Lemma leads us to introduce the spectral projector of A on ker(A), given by Π0= 1 2iπ Z γ (z Id−A)−1dz∈ L(H),
where γ denotes a positively oriented circle centered on 0 with a radius so small that 0 is the single eigenvalue ofA in the interior of γ. We set ˙H = (Id −Π0)H and equip this space with the norm
k(u0, u1)k2H˙ :=|(u0, u1)|2H=kA
1 2u0k2
H+ku1k2H,
and associated inner product. This is indeed a norm on ˙H since k(u0, u1)kH˙ = 0 is equivalent to
(u0, u1)∈ ker(A) × {0} = Π0H.
Besides, we set ˙A = A|H˙ with domain D( ˙A) = D(A) ∩ ˙H. A first remark is that Sp( ˙A) =
Sp(A) \ {0}, so that Sp( ˙A) ∩ iR = ∅.
The remainder of the proof consists in applying Theorem 4.1 to the operator ˙A in ˙H. We first check the assumptions of Theorem 4.1 and describe the solutions of the evolution problem (2.4) (or equivalently (2.3)).
Lemma 4.3. The operator ˙A generates a contraction C0-semigroup on ˙
H, denoted (et ˙A)
t≥0.
More-over, for all initial data U0 ∈ H, Problem (2.4) (or equivalently (2.3)) has a unique solution
U ∈ C0(R+;
H), issued from U0, that can be decomposed as
U (t) = et ˙A(Id−Π0)U0+ Π0U0, for all t≥ 0. (4.4)
As a consequence, we can apply Theorem 4.1 to the semigroup generated by ˙A. The proof of Proposition 2.4 will be achieved when the following lemmata are proved.
Lemma 4.4. Conditions (2.6) and (4.1) are equivalent.
Lemma 4.5. Conditions (2.9) and (2.10) are equivalent. Conditions (2.7) and (2.8) are equivalent. Lemma 4.6. There exist C > 1 and s0> 0 such that for s∈ R, |s| ≥ s0,
k(is Id − ˙A)−1k L( ˙H)− C |s| ≤ k(is Id −A) −1k L(H)≤ k(is Id − ˙A)−1kL( ˙H)+ C |s|, (4.5) and C−1|s|kP (is)−1k
L(H)≤ k(is Id −A)−1kL(H)≤ C 1 + |s|kP (is)−1kL(H)
. (4.6) In particular this implies that (4.2), (2.7) and (2.9) are equivalent.
The proof of Lemma 4.6 is more or less classical and we follow [Leb96, BH07].
Proof of Lemma 4.2. AsA has compact resolvent, its spectrum contains only isolated eigenvalues. Suppose that z∈ Sp(A), then we have, for some (u0, u1)∈ D(A) \ {0},
u1 = zu0,
−Au0− BB∗u1 = zu1,
and in particular
with u0∈ D(A) \ {0}.
Suppose that z ∈ iR, then, this yields Au0− Im(z)2u0+ i Im(z)BB∗u0= 0. Following [Leb96],
taking the inner product of this equation with u0yields i Im(z)kB∗u0k2Y = 0. Hence, either Im(z) = 0,
or B∗u
0 = 0. In the first case, Au0 = 0, i.e. u0 ∈ ker(A), and u1 = 0. This yields ker(A) ⊂
ker(A)×{0} (and the other inclusion is clear). In the second case, u0is an eigenvector of A associated
to the eigenvalue Im(z)2 and satisfies B∗u
0 = 0, which is absurd, according to Assumption (2.5).
Thus, Sp(A) ∩ iR ⊂ {0}.
Now, for a general eigenvalue z ∈ C, taking the inner product of (4.7) with u0 yields
(Au0, u0)H+ (Re(z)2− Im(z)2)ku0k2H+ Re(z)kB∗u0k2Y = 0,
2 Re(z) Im(z)ku0k2H+ Im(z)kB∗u0k2Y = 0.
(4.8) If Im(z)6= 0, then, the second equation of (4.8) together with Sp( ˙A) ∩ iR ⊂ {0} gives
0 > Re(z) =−12kB∗u0k 2 Y ku0k2H ≥ − 1 2kB ∗k2 L(H;Y ).
If Im(z) = 0, then, the first equation of (4.8) together with ( ˙Au0, u0)H≥ 0 gives − Re(z)kB∗u0k2Y ≥
Re(z)2
ku0k2H, which yields
0≥ Re(z) ≥ −kB∗k2 L(H;Y ).
Following [Leb96], we now give the link between P (z)−1 and (z Id−A)−1 for z /∈ Sp(A). Taking
F = (f0, f1)∈ H, and U = (u0, u1), we have
F = (z Id−A)U ⇐⇒
u1= zu0− f0,
P (z)u0= f1+ (BB∗+ z Id)f0. (4.9)
As a consequence, we obtain that P (z) : D(A)→ H is invertible if and only if (z Id −A) : D(A) → H is invertible, i.e. if and only if z /∈ Sp(A). Moreover, for such values of z, System (4.9) is equivalent
to
u0= P (z)−1f1+ P (z)−1(BB∗+ z Id)f0,
u1= zP (z)−1f1+ zP (z)−1(BB∗+ z Id)f0− f0,
which can be rewritten as (4.3). This concludes the proof of Lemma 4.2.
Proof of Lemma 4.3. Let us check that ˙A is a maximal dissipative operator on ˙H [Paz83]. First, it is dissipative since, for U = (u0, u1)∈ D( ˙A),
( ˙AU, U)H˙ = (A 1 2u 1, A 1 2u 0)H− (Au0, u1)H− (BB∗u1, u1)H=−kB∗u1k2Y ≤ 0.
Next, the fact that A − Id is onto is a consequence of Lemma 4.2. Hence, for all F ∈ ˙H ⊂ H, there exists U ∈ D(A) such that (A − Id)U = F . Applying (Id −Π0) to this identity yields ( ˙A −
Id)(Id−Π0)U = F , so that ˙A−Id : D( ˙A) → ˙H is onto. According to the Lumer-Phillips Theorem (see
for instance [Paz83, Chapter 1, Theorem 4.3]) ˙A generates a contraction C0-semigroup on ˙
H. Then, Formula (4.4) directly comes from the linearity of Equation (2.4) (or equivalently (2.3)) together with the decomposition of the initial condition U0= (I− Π0)U0+ Π0U0.
Proof of Lemma 4.4. Condition (4.1) is equivalent to the existence of C > 0 such that for all t > 0, and ˙U0∈ ˙H, we have
ket ˙AA˙−1U˙0kH˙ ≤ C
tαk ˙U0kH˙.
This can be rephrased as
ket ˙AU˙0k ˙
H≤
C
tαk ˙A ˙U0kH˙, (4.10)
for all t > 0, and ˙U0 ∈ D( ˙A). Now, take any U0 = (u0, u1) ∈ D(A), and associated projection
˙
U0= (Id−Π0)U0∈ D( ˙A). According to (4.4), we have
E(u, t) = 1 2 kA 1 2u(t)k2 H+k∂tu(t)k2H = 1 2|e t ˙AU˙0+ Π0U0|2 H= 1 2ke t ˙AU˙0k2 ˙ H,
and
|AU0|H=| ˙A ˙U0+AΠ0U0|H=k ˙A ˙U0kH˙.
This shows that (4.10) is equivalent to (2.6), and concludes the proof of Lemma 4.4.
Proof of Lemma 4.5. First, (2.9) clearly implies (2.10). To prove the converse, for u ∈ D(A), we have
(P (is)u, u)H= (A− s2Id)u, uH+ iskB∗uk2Y.
Taking the imaginary part of this identity gives skB∗uk2
Y = Im(P (is)u, u)H, so that, using the Young
inequality, we obtain for all ε > 0, |s|α1kB∗uk2 Y =|s| 1 α−1| Im(P (is)u, u)H| ≤|s| 2 α−2 4ε kP (is)uk 2 H+ εkuk2H.
Plugging this into (2.10) and taking ε sufficiently small, we obtain that for some C > 0 and s0≥ 0,
for any s∈ R, |s| ≥ s0, kuk2 H≤ C|s| 2 α−2kP (is)uk2 H,
which yields (2.9). Hence (2.9) and (2.10) are equivalent.
Second, Condition (2.8) clearly implies (2.7) and it only remains to prove the converse. For z∈ C, we write r = Re(z) and s = Im(z). We have the identity
((r + is) Id−A)−1= (is Id−A)−1 Id +r(is Id−A)−1−1. (4.11) Hence, assuming kr(is Id −A)−1kL(H)≤ 1 2, (4.12) this gives Id +r(is Id −A)−1−1 L(H)= ∞ X k=0 (−1)k r(is Id−A)−1k L(H) ≤ 2. As a consequence of (4.11) and (2.7), we then obtain
((r + is) Id −A)−1
L(H)≤ 2k(is Id −A)−1kL(H)≤ 2C|s|
1 α,
for all s≥ s0, under Condition (4.12). Finally, (2.7) also yields
kr(is Id −A)−1kL(H)≤ |r|C|s|
1 α,
so that Condition (4.12) is realised as soon as |r| ≤ 1
2C|s|α1. This proves (2.8) and concludes the
proof of Lemma 4.5.
Proof of Lemma 4.6. To prove (4.5), we first remark that the norms k · kH˙ and k · kH are
equiv-alent on ˙H, so that the norms k · kL( ˙H) and k · kL(H) are equivalent on L( ˙H). Next, we have (is Id− ˙A)−1(Id−Π0) = (is Id−A)−1(Id−Π0) and
k(is Id − ˙A)−1k
L(H)=k(is Id − ˙A)−1(Id−Π0)kL(H)=k(is Id −A)−1(Id−Π0)kL(H)
≤ k(is Id −A)−1kL(H)+k(is Id −A)−1Π0kL(H),
together with
k(is Id −A)−1k
L(H)=k(is Id − ˙A)−1(Id−Π0) + (is Id−A)−1Π0kL(H)
Moreover, for|s| ≥ 1, we have
k(is Id −A)−1Π0kL(H)=k(is)−1Π0kL(H)=
1
|s|kΠ0kL(H)= C |s|, which concludes the proof of (4.5).
Let us now prove (4.6). For concision, we set H1 = D(A
1
2) endowed with the graph norm
kukH1 = k(A + Id) 1
2ukH and denote by H
−1 = D(A
1
2)′ its dual space. The operator A can be
uniquely extended as an operatorL(H1; H−1), still denoted A fo simplicity. With this notation, the
space H−1 can be equipped with the natural normkukH−1=k(A + Id)− 1 2ukH.
As a consequence of Formula (4.3), and using the fact that Sp(A)∩iR ⊂ {0}, there exist constants C > 1 and s0> 0 such that for all s∈ R, |s| ≥ s0,
C−1M (s)≤ k(is Id −A)−1k
L(H)≤ CM(s) (4.13)
with
M (s) =kP (is)−1(BB∗+ is Id)k
L(H1)+kP (is)−1kL(H;H1)
+kP (is)−1(isBB∗− s2Id)
− Id kL(H1;H)+ksP (is)−1kL(H)
(4.14) On the one hand, this direcly yields for s∈ R, |s| ≥ s0,
|s|kP (is)−1k
L(H) ≤ Ck(is Id −A)−1kL(H).
This proves that (4.2) implies (2.9).
On the other hand, we have to estimate each term of (4.14). First, using Au = P (is)u + s2u−
isBB∗u, we have
kuk2 H1=kA
1 2uk2
H+kuk2H= P (is)u + s2u− isBB∗u, u
H+kuk 2 H = Re P (is)u, uH+ (s2+ 1) kuk2 H≤ C kP (is)uk2H+ (s2+ 1)kuk2H ≤ C1 + (s2+ 1)kP (is)−1k2 L(H) kP (is)uk2 H, so that kP (is)−1k L(H;H1)≤ C 1 + (|s| + 1)kP (is)−1kL(H) . (4.15)
Second, the same computation for (P (is)−1)∗= (A− s2Id
−isBB∗)−1(the adjoint of P (is)−1 in
the space H) in place of P (is)−1 leads to (P (is)−1)∗∈ L(H; H
1), together with the estimate
k(P (is)−1)∗kL(H;H1)≤ C 1 + (|s| + 1)kP (is)
−1k L(H).
By transposition, we havet(P (is)−1)∗∈ L(H
−1; H), together with the estimate
kt(P (is)−1)∗kL(H−1;H)≤ k(P (is) −1)∗k L(H;H1)≤ C 1 + (|s| + 1)kP (is) −1k L(H) . (4.16) Moreover,t(P (is)−1)∗ is defined, for every u∈ H, v ∈ H
−1, by t(P (is)−1)∗v, u H = v, (P (is)−1)∗u H− 1,H1 = (A + Id)−12v, (A + Id) 1 2(P (is)−1)∗u H.
In particular, taking v∈ H gives
t(P (is)−1)∗v, u
H= P (is)−1v, u
H,
which implies that the restriction of the operator t(P (is)−1)∗ to H coincides with P (is)−1. For
Equation (4.16) can thus be rewritten kP (is)−1k
L(H−1;H)≤ C 1 + (|s| + 1)kP (is)−1kL(H)
. (4.17)
Then, we have P (is)−1(isBB∗− s2Id)
− Id = P (is)−1A, so that
kP (is)−1(isBB∗− s2Id)− Id kL(H1;H)=kP (is)
−1Ak L(H1;H)≤ kP (is) −1k L(H−1;H)kAkL(H1;H−1) ≤ 1 + (|s| + 1)kP (is)−1k L(H) (4.18) Third, for |s| ≥ 1 we write
P (is)−1(BB∗+ is Id) = i s P (is)
−1A− Id, (4.19)
and it remains to estimate the termkP (is)−1Ak
L(H1)in (4.14). For f ∈ H1, we set u = P (is)−1Af .
We have u∈ H1, together with
(A− s2Id +isBB∗)u = Af.
Taking the real part of the inner product of this identity with u, we find kA12uk2
H− s2kuk2H= Re(Af, u)H ≤ kAfkH−1kukH1≤ CkfkH1kukH1,
as A∈ L(H1, H−1). Hence
kuk2H1 ≤ C(1 + s
2)
kuk2H+ Ckfk2H1
Using (4.17), this gives kuk2 H1 ≤ C(1 + s 2) kP (is)−1Ak2 L(H1;H)kfk 2 H1+ Ckfk 2 H1 ≤ C(1 + s2) kP (is)−1k2 L(H−1;H)kfk 2 H1+ Ckfk 2 H1 ≤ C(1 + s2) 1 + (|s| + 1)kP (is)−1kL(H)2kfk2H1,
and finallykP (is)−1Ak
L(H1) ≤ C(1 + |s|) 1 + (|s| + 1)kP (is)−1kL(H)
. Coming back to (4.19), we have, for|s| ≥ 1,
kP (is)−1(BB∗+ is Id)kL(H1)≤ C 1 + |s|kP (is)
−1k
L(H). (4.20)
Finally, combining (4.15), (4.18) and (4.20), together with (4.13)-(4.14), we obtain for|s| ≥ 1, k(is Id −A)−1kL(H)≤ C 1 + |s|kP (is)−1kL(H)
. This concludes the proof of Lemma 4.6.
Part III
Proof of Theorem 2.6: smooth damping
coefficients on the torus
To prove Theorem 2.6, we shall instead prove Estimate (2.9) with α = 1+δ1 (which, according to Proposition 2.4, is equivalent to the statement of Theorem 2.6). Let us first recast (2.9) with α = 1
1+δ
in the semiclassical setting : taking h = s−1, we are left to prove that there exist C > 1 and h 0> 0
such that for all h≤ h0, for all u∈ H2(T2), we have
We prove this inequality by contradiction, using the notion of semiclassical measures. The idea of developing such a strategy for proving energy estimates, together with the associate technology, originates from Lebeau [Leb96].
We assume that (4.21) is not satisfied, and will obtain a contradiction at the end of Section 11. Hence, for all n∈ N, there exists 0 < hn≤ 1n and un ∈ H2(T2) such that
kunkL2(T2)> n hδ nkP (i/h n)unkL2(T2). Setting vn = un/kunkL2(T2), and Phn b =−h 2 n∆− 1 + ihnb(x) = h2nP (i/hn), we then have, as n→ ∞, hn→ 0+, kvnkL2(T2)= 1, h−2−δ n kPbhnvnkL2(T2)→ 0.
Our goal is now to associate to the sequence (un, hn) a semiclassical measure on the cotangent bundle
µ on T∗T2 = T2× (R2)∗ (where (R2)∗ is the dual space of R2). To obtain a contradiction, we shall prove both that µ(T∗T2) = 1, and that µ = 0 on T∗T2.
From now on, we drop the subscript n of the sequences above, and write h in place of hn and vh
in place of vn. We study sequences (h, vh) such that h→ 0+ and
(
kvhkL2(T2)= 1
kPh
bvhkL2(T2)= o(h2+δ), as h→ 0+.
(4.22) In particular, this last equation also yields the key information
(bvh, vh)L2(T2)= h−1Im(Pbhvh, vh)L2(T2)= o(h1+δ), as h→ 0+.
In the following, it will be convenient to identify (R2)∗ and R2 through the usual inner product.
In particular, the cotangent bundle T∗T2= T2× (R2)∗ will be identified with T2× R2.
5
Semiclassical measures
We denote by T∗T2the compactification of T∗T2obtained by adding a point at infinity to each fiber (i.e., the set T2
× (R2
∪ {∞})). A neighbourhood of (x, ∞) ∈ T∗T2 is a set U × {∞} ∪ R2
\ K, where U is a neighbourhood of x in T2and K a compact set in R2. Endowed with this topology, the
set T∗T2is compact.
We denote by S0(T∗T2), S0 for short, the space of functions a(x, ξ) that satisfy the following properties:
1. a∈ C∞(T∗T2).
2. There exists a compact set K ⊂ R2 and a constant k
0 ∈ C such that a(x, ξ) = k0 for all
ξ∈ R2
\ K.
Note that we have in particular Cc∞(T∗T2)⊂ S0(T∗T2).
To a symbol a ∈ S0(T∗T2), we associate its semiclassical Weyl quantization Oph(a) by For-mula (A.1), which, according to the Calder´on-Vaillancourt Theorem (see Appendix A) defines a uniformly bounded operator on L2(T2).
From the sequence (vh, h) (see for instance [GL93]), we can define (using again the
Calder´on-Vaillancourt Theorem) the associated Wigner distribution Vh
∈ (S0)′ by
Decomposing vhand a in Fourier series, ˆ vh(k) = 1 2π Z T2 e−ik·xvh(x)dx, a(h, k, ξ) =ˆ 1 2π Z T2 e−ik·xa(h, x, ξ)dx,
the expression (5.1) can be more explicitly rewritten as Vh, a (S0)′ ,S0 = 1 2π X k,j∈Z2 ˆ a h, j− k,h 2(k + j) ˆ vh(k)ˆvh(j).
Proposition 5.1. The family (Vh) is bounded in (S0)′. Hence, there exists a subsequence of the
sequence (h, vh) and an element µ∈ (S0)′, such that Vh⇀ µ weakly in (S0)′, i.e.
(Oph(a)vh, vh)L2(T2)→ hµ, ai(S0)′
,S0 for all a∈ S0(T∗T2). (5.2)
In addition,hµ, ai(S0)′,S0 is nonnegative if a is; in other words, µ may be identified with a nonnegative
Radon measure on T∗T2.
Notation: in what follows we shall denote byM+(T∗T2) the set of nonnegative Radon measures
on T∗T2.
Proof. The proof is an adaptation from the original proof of G´erard [G´er91] (see also [GL93] in the semiclassical setting).
The fact that the Wigner distributions Vh are uniformly bounded in (S0)′ follows from the Calder´on-Vaillancourt theorem (see Appendix A), and from the boundedness of (vh) in L2(T2).
The sharp G˚arding inequality gives the existence of C > 0 such that, for all a≥ 0 and h > 0, (Oph(a)vh, vh)L2(T2)≥ −Chkvhk2L2(T2),
so that the distribution µ is nonnegative (and is hence a measure).
6
Zero-th and first order informations on µ
To simplify the notation, we setPbh= P0h+ ihb(x), with P0h=−h2∆− 1 = Oph(|ξ|2− 1).
The geodesic flow on the torus φτ : T∗T2 → T∗T2 for τ ∈ R is the flow generated by the
Hamiltonian vector field associated to the symbol 12(|ξ|2− 1), i.e. by the vector field ξ · ∂
xon T∗T2.
Explicitely, we have
φτ(x, ξ) = (x + τ ξ, ξ), τ∈ R, (x, ξ) ∈ T∗T2.
Note that φτpreserves the ξ-component, and, in particular every energy layer{|ξ|2= C > 0} ⊂ T∗T2.
Now, we describe the first properties of the measure µ implied by (4.22).
We recall that for ν ∈ D′(T∗T2), (φτ)∗ν ∈ D′(T∗T2) is defined byh(φτ)∗ν, ai = hν, a ◦ φτi for all a ∈ C∞
c (T∗T2). In particular, (φτ)∗ν is a measure if ν is. We shall say that ν is an invariant
measure if it is invariant by the geodesic flow, i.e. (φτ)∗ν = ν for all τ ∈ R.
Proposition 6.1. Let µ be as in Proposition 5.1. We have 1. supp(µ)⊂ {|ξ|2= 1
} (hence is compact in T∗T2), 2. µ(T∗T2) = 1,
3. µ is invariant by the geodesic flow, i.e. (φτ)∗µ = µ,
4. hµ, biMc(T∗T2),C0(T∗T2) = 0, where Mc(T∗T2) denotes the space of compactly supported
In other words, µ is an invariant probability measure on T∗T2 vanishing on {b > 0}.
These are standard arguments, that we reproduce here for the reader’s comfort. In particular, we recover all informations required to prove the Bardos-Lebeau-Rauch-Taylor uniform stabilization theorem under GCC. But we do not use here the second order informations of (4.22); this will be the key point to prove Theorem 2.6.
Proof. First, we take χ∈ C∞(T∗T2) depending only on the ξ variable, such that χ≥ 0, χ(ξ) = 0
for|ξ| ≤ 2, and χ(ξ) = 1 for |ξ| ≥ 3. Hence, |ξ|χ(ξ)2−1 ∈ C∞(T∗T2) and we have the exact composition
formula Oph(χ) = Oph χ(ξ) |ξ|2− 1 P0h,
since both operators are Fourier multipliers. Moreover, Oph
χ(ξ) |ξ|2−1 is a bounded operator on L2(T2). As a consequence, we have Vh, χ (S0)′,S0 → hµ, χiM(T∗T2),C0(T∗T2), together with Vh, χ (S0)′ ,S0 = Oph χ(ξ) |ξ|2− 1 P0hvh, vh L2(T2) = Oph χ(ξ) |ξ|2− 1 Pbhvh, vh L2(T2) − ih Oph χ(ξ) |ξ|2− 1 bvh, vh L2(T2) . Since kPh
bvhkL2(T2) = o(1) and kvhkL2(T2) = 1, both terms in this expression vanish in the limit
h→ 0+. This implies that
hµ, χiM(T∗T2),C0(T∗T2)= 0. Since this holds for all χ as above, we have
supp(µ)⊂ {|ξ|2= 1
}, which proves Item 1.
In particular, this implies that µT∗T2\ T∗T2= 0. Now, Item 2 is a direct consequence of 1 =
kvhk2L2(T2)→ hµ, 1iM(T∗T2),C0(T∗T2) and Item 1. Item 4 is a direct consequence of (bvh, vh)L2(T2)=
o(1). Finally, for a∈ C∞ c (T∗T2), we recall that P0h, Oph(a) =h i Oph({|ξ| 2 − 1, a}) = 2hi Oph(ξ· ∂xa),
is a consequence of the Weyl quantization (any other quantization would have left an error term of orderO(h2)). Hence, (5.1) yields
Vh, ξ· ∂xa D′ (T∗T2),C∞ c (T ∗T2)→ hµ, ξ · ∂xaiM(T∗T2),C0 c(T ∗T2), (6.1) together with Vh, ξ· ∂xa D′(T∗T2),C∞ c (T∗T2)= i 2h P0h, Oph(a) vh, vhL2(T2) = i 2h Oph(a)vh, P h 0vhL2(T2)− i 2h Oph(a)P h 0vh, vhL2(T2) = i 2h Oph(a)vh, P h bvhL2(T2)− i 2h Oph(a)P h bvh, vhL2(T2) −12(Oph(a)vh, bvh)L2(T2)− 1 2(Oph(a)bvh, vh)L2(T2). (6.2)
In this expression, we have 1h Oph(a)vh, PbhvhL2(T2)→ 0 and
1
h Oph(a)Pbhvh, vh
L2(T2)→ 0 since
kPh
bvhkL2(T2)= o(h). Moreover, the last two terms can be estimated by
| (Oph(a)bvh, vh)L2(T2)| ≤ k
√
bvhkL2(T2)k
√
since (bvh, vh)L2(T2) = o(1). This yields Vh, ξ· ∂ xa D′(T∗T2),C∞ c (T∗T2) → 0, so that, using (6.1), hµ, ξ · ∂xaiM(T∗T2),C0 c(T∗T2) = 0 for all a∈ C ∞
c (T∗T2). Replacing a by a◦ φτ and integrating with
respect to the parameter τ gives (φτ)∗µ = µ, which concludes the proof of Item 3.
7
Geometry on the torus and decomposition of invariant
mea-sures
7.1
Resonant and non-resonant vectors on the torus
In this section, we collect several facts concerning the geometry of T∗T2and its resonant subspaces. Most of the setting and the notation comes from [AM11, Section 2].
We shall say that a submodule Λ⊂ Z2 is primitive if
hΛi ∩ Z2= Λ, where
hΛi denotes the linear subspace of R2 spanned by Λ. The family of all primitive submodules will be denoted byP.
Let us denote by Ωj⊂ R2, for j = 0, 1, 2, the set of resonant vectors of order exactly j, i.e.,
Ωj :={ξ ∈ R2such that rk(Λξ) = 2− j}, with Λξ :=k∈ Z2 such that ξ· k = 0 = ξ⊥∩ Z2.
Note that the sets Ωj form a partition of R2, and that we have
• Ω0={0};
• ξ ∈ Ω1if and only if the geodesic issued from any x∈ T2 in the direction ξ is periodic;
• ξ ∈ Ω2if and only if the geodesic issued from any x∈ T2 in the direction ξ is dense in T2.
For each Λ∈ P such that rk(Λ) = 1, we define Λ⊥:=ξ∈ R2 such that ξ
· k = 0 for all k ∈ Λ , TΛ:=hΛi /2πΛ,
TΛ⊥ := Λ⊥/(2πZ2∩ Λ⊥).
Note that TΛ and TΛ⊥ are two submanifolds of T2 diffeomorphic to one-dimensional tori. Their
cotangent bundles admit the global trivialisations T∗TΛ= TΛ× hΛi and T∗TΛ⊥ = TΛ⊥× Λ⊥.
For a function f on T2 with Fourier coefficients ( ˆf (k))
k∈Z2, and Λ∈ P, we shall say that f has
only Fourier modes in Λ if ˆf (k) = 0 for k /∈ Λ. This means that f is constant in the direction Λ⊥,
or, equivalently, that σ· ∂xf = 0 for all σ ∈ Λ⊥. We denote by LpΛ(T2) the subspace of Lp(T2)
consisting of functions having only Fourier modes in Λ. For a function f ∈ L2(T2) (resp. a symbol
a∈ S0(T∗T2)), we denote byhfi
Λ its orthogonal projection on L2Λ(T2), i.e. the average of f along
Λ⊥: hfiΛ(x) := X k∈Λ eik·x 2π f (k)ˆ resp. haiΛ(x, ξ) := X k∈Λ eik·x 2π ˆa(k, ξ) ! . If rk(Λ) = 1 and v is a vector in Λ⊥\ {0}, we also have
hfiΛ(x) = lim T →∞ 1 T Z T 0 f (x + tv)dt. (7.1) In particular, note that hfiΛ (resp.haiΛ) is nonnegative if f (resp. a) is, and thathfiΛ ∈ C∞(T2)
(resp.haiΛ∈ S0(T∗T2)) if f∈ C∞(T2) (resp. a∈ S0(T∗T2)). Finally, given f ∈ L∞
Λ(T2), we denote by mf the bounded operator on L2Λ(T2), consisting in the
7.2
Decomposition of invariant measures
We denote by M+(T∗T2) the set of finite, nonnegative measures on T∗T2. With the definitions above, we have the following decomposition Lemmata, proved in [Mac10] or [AM11, Section 2]. These properties are given for general measures µ∈ M+(T∗T2). Of course, they apply in particular to the measure µ defined by Proposition 5.1.
Lemma 7.1. Let µ∈ M+(T∗T2). Then µ decomposes as a sum of nonnegative measures µ = µ|T2×{0}+ µ|T2×Ω 2+ X Λ∈P,rk(Λ)=1 µ|T2×(Λ⊥ \{0}) (7.2)
Given µ∈ M+(T∗T2), we define its Fourier coefficients by the complex measures on R2: ˆ µ(k,·) := Z T2 e−ik·x 2π µ(dx,·), k∈ Z.
One has, in the sense of distributions, the following Fourier inversion formula: µ(x, ξ) = X
k∈Z2
eik·x
2π µ(k, ξ).ˆ Lemma 7.2. Let µ∈ M+(T∗T2) and Λ∈ P. Then, the distribution
hµiΛ(x, ξ) :=
X
k∈Λ
eik·x
2π µ(k, ξ),ˆ is inM+(T∗T2) and satisfies, for all a∈ Cc∞(T∗T2),
hhµiΛ, aiM(T∗T2),C0 c(T
∗T2)=hµ, haiΛiM(T∗T2),C0 c(T
∗T2).
Lemma 7.3. Let µ∈ M+(T∗T2) be an invariant measure. Then, for all Λ∈ P, µ|T2×(Λ⊥
\{0}) is
also a nonnegative invariant measure and µ|T2×(Λ⊥
\{0})=hµiΛ|T2×(Λ⊥
\{0}).
Let us now come back to the measure µ given by Proposition 5.1, which satisfies all properties listed in Proposition 6.1. In particular, this measure vanishes on the non-empty open subset of T2
given by{b > 0} (see Item 4 in Proposition 6.1). As a consequence of Proposition 6.1, and of the three lemmata above, this yields the following lemma.
Lemma 7.4. We have µ =PΛ∈P,rk(Λ)=1µ|T2×(Λ⊥
\{0}).
As a consequence of Proposition 6.1, we have indeed that the measure µ is supported in{|ξ| = 1}, which implies µ|T2×{0}= 0. In addition, Lemma 7.3 applied with Λ ={0} implies that µ|T2×Ω
2 is
constant in x – and thus vanishes everywhere since it vanishes on{b > 0}.
Remark 7.5. Since the measure µ is supported in{|ξ| = 1} (Proposition 6.1, Item 1), we have µ|T2×Λ⊥ = µ|T2×(Λ⊥
\{0})
(which simplifies the notation).
As a consequence of these lemmata and the last remark, the study of the measure µ is now reduced to that of all nonnegative invariant measures µ|T2×Λ⊥ with rk(Λ) = 1.
The aim of the next sections is to prove that the measure µ|T2×Λ⊥ vanishes identically, for each
7.3
Geometry of the subtori T
Λand T
Λ⊥To study the measure µ|T2×(Λ⊥
\{0}), we need to describe briefly the geometry of the subtori TΛand
TΛ⊥ of T2, and introduce adapted coordinates.
We define χΛthe linear isomorphism
χΛ : Λ⊥× hΛi → R2: (s, y)7→ s + y,
and denote by ˜χΛ: T∗Λ⊥× T∗hΛi → T∗R2 its extension to the cotangent bundle. This application
can be defined as follows: for (s, σ)∈ T∗Λ⊥= Λ⊥× (Λ⊥)∗ and (y, η)∈ T∗hΛi = hΛi × hΛi∗, we can
extend σ to a covector of R2vanishing on
hΛi and η to a covector of R2vanishing on Λ⊥. Remember
that we identify (R2)∗with R2through the usual inner product; thus we can also see σ as an element
of Λ⊥ and η as an element ofhΛi. Then, we have
˜
χΛ(s, σ, y, η) = (s + y, σ + η)∈ T∗R2= R2× (R2)∗.
Conversely, any ξ ∈ (R2)∗ can be decomposed into ξ = σ + η where σ ∈ Λ⊥ and η ∈ hΛi. We
denote by PΛ the orthogonal projection of R2ontohΛi, i.e. PΛξ = η.
Next, the map χΛ goes to the quotient, giving a smooth Riemannian covering of T2by
πΛ: TΛ⊥× TΛ→ T2: (s, y)7→ s + y.
We shall denote by ˜πΛ its extension to cotangent bundles:
˜
πΛ: T∗TΛ⊥× T∗TΛ→ T∗T2.
As the map πΛ is not an injection (because the torus TΛ⊥ × TΛ contains several copies of T2), we
introduce its degree pΛ, which is also equal to
Vol(TΛ⊥×TΛ)
Vol(T2) .
Then, the application
TΛu := √1
pΛ
u◦ χΛ,
defines a linear isomorphism L2
loc(R2) → L2loc(Λ⊥× hΛi). Note that because of the factor √1pΛ,
TΛ maps L2(T2) isometrically into a subspace of L2(TΛ⊥ × TΛ). Moreover, TΛ maps L2
Λ(T2) into
L2(T
Λ) ⊂ L2(TΛ⊥ × TΛ), since the nonvanishing Fourier modes of u∈ L2
Λ(T2) correspond only to
frequencies k∈ Λ. This reads
TΛu(s, y) = √1
pΛu(y) for (s, y)∈ TΛ
⊥× TΛ. (7.3)
Since ˜χΛ is linear, we have, for any a∈ C∞(T∗R2)
TΛOph(a) = Oph(a◦ ˜χΛ)TΛ, (7.4)
where on the left Oph is the Weyl quantization on R2 (A.1), and on the right Op
h is the Weyl
quantization on Λ⊥ × hΛi. Next, we denote by OpΛ⊥
h and OpΛh the Weyl quantization operators
defined on smooth test functions on T∗Λ⊥× T∗hΛi and acting only on the variables in T∗Λ⊥ and
T∗hΛi respectively, leaving the other frozen. For any a ∈ C∞
c (T∗Λ⊥× T∗hΛi), we have : Oph(a) = OpΛ ⊥ h ◦ OpΛh(a) = OpΛh◦ OpΛ ⊥ h (a). (7.5)
Now, if the symbol a∈ C∞
c (T∗T2) has only Fourier modes in Λ, we remark, in view of (7.3), that
a◦ ˜πΛ does not depend on s∈ TΛ⊥. Therefore, we sometimes write a◦ ˜πΛ(σ, y, η) for a◦ ˜πΛ(s, σ, y, η)
and (7.4)-(7.5) give
TΛOph(a) = OpΛh◦ OpΛ
⊥
h (a◦ ˜πΛ)TΛ= OpΛh(a◦ ˜πΛ(hDs,·, ·))TΛ. (7.6)
Note that for every σ∈ Λ⊥, the operator OpΛ
h(a◦ ˜πΛ(σ,·, ·)) maps L2(TΛ) into itself. More precisely,
8
Change of quasimode and construction of an invariant
cut-off function
In this section, we first construct from the quasimode vh another quasimode wh, that will be easier
to handle when studying the measure µ|T2×Λ⊥. Indeed wh is basically a microlocalization of vh in
the direction Λ⊥ at a precise concentration rate.
Moreover, we introduce a cutoff function χΛ
h(x) = χΛh(y, s), well-adapted to the damping
coeffi-cient b and to the invariance of the measure µ|T2×Λ⊥ in the direction Λ⊥ (this cutoff function plays
the role of the function χ(b/h) used in [BH07] in the case where b is itself invariant in the direction Λ⊥). Its construction is a key point in the proof of Theorem 2.6.
Let χ∈ C∞
c (R) be a nonnegative function such that χ = 1 in a neighbourhood of the origin. We
first define wh:= Oph χ |PΛξ| hα vh,
which implicitely depends on α∈ (0, 1). The following lemma implies that, for δ and α sufficiently small, wh is as well a o(h2+δ)-quasimode for Pbh.
Lemma 8.1. For any α > 0 such that δ +ε 2+ α≤ 1 2, 3α + 2δ < 1, (8.1) we have kPbhwhkL2(T2)= o(h2+δ).
As a consequence of this lemma, the semiclassical measures associated to whsatisfy in particular
the conclusions of Proposition 6.1. Moreover, the following proposition implies that the sequence wh
contains all the information in the direction Λ⊥.
Proposition 8.2. For any a ∈ C∞
c (T∗T2) and any α ∈ (0, 3/4) satisfying the assumptions of
Lemma 8.1, we have
hµ|T2×Λ⊥, ai
M(T∗T2),C0
c(T∗T2)= limh→0(Oph(a)wh, wh)L2(T2).
Next, we state the desired properties of the cutoff function χΛ
h. The proof of its existence is a
crucial point in the proof of Theorem 2.6.
Proposition 8.3. For δ = 8ε, and ε < 761, there exists α satisfying (8.1), such that for any constant c0> 0, there exists a cutoff function χΛh ∈ C∞(T2) valued in [0, 1], such that
1. χΛ
h = χΛh(y) does not depend on the variable s (i.e. χΛh is Λ⊥-invariant),
2. k(1 − χΛ h)whkL2(T2)= o(1), 3. b≤ c0h on supp(χΛh), 4. k∂yχΛhwhkL2(T2)= o(1), 5. k∂2 yχΛhwhkL2(T2)= o(1).
Note that the function χΛ
h implicitely depends on the constant c0, that will be taken arbitrarily
small in Section 10.
In the particular case where the damping function b is invariant in one direction, this proposition is not needed. In this case, one can take as in [BH07] χΛ
h = χ(c0bh). In the d-dimensional torus, this
cutoff functions works as well if b is invariant in d− 1 directions, and an analogue of Theorem 2.6 can be stated in this setting. Unfortunately, our construction of the function χΛ
h (see the proof of
Proposition 8.3 in Section 13) strongly relies on the fact that all trapped directions are periodic, and fails in higher dimensions.
We give here a proof of Lemma 8.1. Because of their technicality, we postpone the proofs of Propositions 8.2 and 8.3 to Sections 12 and 13 respectively.
